Abstract

Greek ladder is a technique for approximating Cn by rational numbers where n is a positive integer and C is a positive real number. For the classical Greek ladder, the value isC. Based on the continued fraction theory and algebraic equation, the classical Greek ladder in a special case can be reduced to the generalized Fibonacci sequence. By means of proper switching and binary, ternary or quaternary phase modulation, here we have successfully designed the various kinds of nano-photonic devices to produce three-dimensional array foci whose focusing properties satisfy the above mathematical characteristics. With this technology, the diffraction-limited array foci are freely designed or distributed under the requirement at the desired multiple focal planes.

© 2015 Optical Society of America

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References

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    [Crossref]
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2015 (3)

2014 (2)

V. Ferrando, A. Calatayud, P. Andrés, R. Torroba, W. D. Furlan, and J. A. Monsoriu, “Imaging properties of Kinoform Fibonacci lenses,” IEEE Photonics J. 6(1), 6500106 (2014).
[Crossref]

C. Horgan and K. Herzinger, “A further investigation of using Theon’s ladder to find roots of quadratic equations,” Int. J. Math. Educ. Sci. Technol. 45(1), 150–158 (2014).
[Crossref]

2013 (3)

2012 (1)

2011 (1)

2010 (2)

2009 (2)

2008 (1)

H.-H. Chung, N. M. Bradman, M. R. Davidson, and P. H. Holloway, “Dual wavelength photon sieves,” Opt. Eng. (Bellingham) 47(11), 118001 (2008).
[Crossref]

2006 (2)

2005 (2)

G. Andersen, “Large optical photon sieve,” Opt. Lett. 30(22), 2976–2978 (2005).
[Crossref] [PubMed]

T. J. Osler, M. Wright, and M. Orchard, “Theon’s ladder for any root,” Int. J. Math. Educ. Sci. Technol. 36(4), 389–398 (2005).
[Crossref]

2003 (3)

2001 (1)

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft X-rays with photon sieves,” Nature 414(6860), 184–188 (2001).
[Crossref] [PubMed]

1995 (1)

1991 (1)

J. A. Sun and A. Cai, “Archaic focusing properties of Fresnel zone plates,” J. Opt. Soc. Am. 8(1), 33–35 (1991).
[Crossref]

1986 (1)

J. R. Ridenhour, “Ladder Approximations of Irrational Numbers,” Math. Mag. 59(2), 95–105 (1986).
[Crossref]

1985 (1)

1977 (1)

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta (Lond.) 24(4), 505–515 (1977).
[Crossref]

1973 (1)

M. D. Tipton and J. E. Dowdey, “Coded aperture imaging with on-axis Fresnel zone plates,” Opt. Eng. 12(5), 166–168 (1973).
[Crossref]

1968 (1)

1967 (1)

Adelung, R.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft X-rays with photon sieves,” Nature 414(6860), 184–188 (2001).
[Crossref] [PubMed]

Andersen, G.

Andrés, P.

V. Ferrando, A. Calatayud, P. Andrés, R. Torroba, W. D. Furlan, and J. A. Monsoriu, “Imaging properties of Kinoform Fibonacci lenses,” IEEE Photonics J. 6(1), 6500106 (2014).
[Crossref]

J. A. Monsoriu, A. Calatayud, L. Remón, W. D. Furlan, G. Saavedra, and P. Andrés, “Bifocal Fibonacci diffraction lenses,” IEEE Photonics J. 5(3), 3400106 (2013).
[Crossref]

Berndt, R.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft X-rays with photon sieves,” Nature 414(6860), 184–188 (2001).
[Crossref] [PubMed]

Bradman, N. M.

H.-H. Chung, N. M. Bradman, M. R. Davidson, and P. H. Holloway, “Dual wavelength photon sieves,” Opt. Eng. (Bellingham) 47(11), 118001 (2008).
[Crossref]

Cai, A.

J. A. Sun and A. Cai, “Archaic focusing properties of Fresnel zone plates,” J. Opt. Soc. Am. 8(1), 33–35 (1991).
[Crossref]

Calatayud, A.

V. Ferrando, A. Calatayud, P. Andrés, R. Torroba, W. D. Furlan, and J. A. Monsoriu, “Imaging properties of Kinoform Fibonacci lenses,” IEEE Photonics J. 6(1), 6500106 (2014).
[Crossref]

J. A. Monsoriu, A. Calatayud, L. Remón, W. D. Furlan, G. Saavedra, and P. Andrés, “Bifocal Fibonacci diffraction lenses,” IEEE Photonics J. 5(3), 3400106 (2013).
[Crossref]

V. Ferrando, A. Calatayud, F. Giménez, W. D. Furlan, and J. A. Monsoriu, “Cantor dust zone plates,” Opt. Express 21(3), 2701–2706 (2013).
[Crossref] [PubMed]

A. Calatayud, V. Ferrando, L. Remón, W. D. Furlan, and J. A. Monsoriu, “Twin axial vortices generated by Fibonacci lenses,” Opt. Express 21(8), 10234–10239 (2013).
[Crossref] [PubMed]

Calvo, M. L.

Cao, H.

Cao, Q.

Cao, W.

Champagne, E.

Cheng, G.

Chung, H.-H.

H.-H. Chung, N. M. Bradman, M. R. Davidson, and P. H. Holloway, “Dual wavelength photon sieves,” Opt. Eng. (Bellingham) 47(11), 118001 (2008).
[Crossref]

Craven, J. M.

Dammann, H.

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta (Lond.) 24(4), 505–515 (1977).
[Crossref]

Davidson, M. R.

H.-H. Chung, N. M. Bradman, M. R. Davidson, and P. H. Holloway, “Dual wavelength photon sieves,” Opt. Eng. (Bellingham) 47(11), 118001 (2008).
[Crossref]

Davis, J. A.

Dong, X.

Dowdey, J. E.

M. D. Tipton and J. E. Dowdey, “Coded aperture imaging with on-axis Fresnel zone plates,” Opt. Eng. 12(5), 166–168 (1973).
[Crossref]

Du, C.

Ferrando, V.

Furlan, W. D.

Giménez, F.

Harm, S.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft X-rays with photon sieves,” Nature 414(6860), 184–188 (2001).
[Crossref] [PubMed]

Herzinger, K.

C. Horgan and K. Herzinger, “A further investigation of using Theon’s ladder to find roots of quadratic equations,” Int. J. Math. Educ. Sci. Technol. 45(1), 150–158 (2014).
[Crossref]

Holloway, P. H.

H.-H. Chung, N. M. Bradman, M. R. Davidson, and P. H. Holloway, “Dual wavelength photon sieves,” Opt. Eng. (Bellingham) 47(11), 118001 (2008).
[Crossref]

Horgan, C.

C. Horgan and K. Herzinger, “A further investigation of using Theon’s ladder to find roots of quadratic equations,” Int. J. Math. Educ. Sci. Technol. 45(1), 150–158 (2014).
[Crossref]

Hu, C.

Hu, J.

Jahns, J.

Jia, J.

J. Jia and C. Xie, “Phase zone photon sieve,” Chin. Phys. B 18(1), 183–188 (2009).
[Crossref]

C. Zhou, J. Jia, and L. Liu, “Circular Dammann grating,” Opt. Lett. 28(22), 2174–2176 (2003).
[Crossref] [PubMed]

Jia, W.

Johnson, R. L.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft X-rays with photon sieves,” Nature 414(6860), 184–188 (2001).
[Crossref] [PubMed]

Ke, J.

Kipp, L.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft X-rays with photon sieves,” Nature 414(6860), 184–188 (2001).
[Crossref] [PubMed]

Klotz, E.

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta (Lond.) 24(4), 505–515 (1977).
[Crossref]

Kyuragi, H.

Liu, L.

Liu, M.

Ma, J.

Mirzaie, S.

Mittra, R.

Monsoriu, J. A.

Orchard, M.

T. J. Osler, M. Wright, and M. Orchard, “Theon’s ladder for any root,” Int. J. Math. Educ. Sci. Technol. 36(4), 389–398 (2005).
[Crossref]

Osler, T. J.

T. J. Osler, M. Wright, and M. Orchard, “Theon’s ladder for any root,” Int. J. Math. Educ. Sci. Technol. 36(4), 389–398 (2005).
[Crossref]

Pons, A.

Remón, L.

J. A. Monsoriu, A. Calatayud, L. Remón, W. D. Furlan, G. Saavedra, and P. Andrés, “Bifocal Fibonacci diffraction lenses,” IEEE Photonics J. 5(3), 3400106 (2013).
[Crossref]

A. Calatayud, V. Ferrando, L. Remón, W. D. Furlan, and J. A. Monsoriu, “Twin axial vortices generated by Fibonacci lenses,” Opt. Express 21(8), 10234–10239 (2013).
[Crossref] [PubMed]

Ridenhour, J. R.

J. R. Ridenhour, “Ladder Approximations of Irrational Numbers,” Math. Mag. 59(2), 95–105 (1986).
[Crossref]

Saavedra, G.

J. A. Monsoriu, A. Calatayud, L. Remón, W. D. Furlan, G. Saavedra, and P. Andrés, “Bifocal Fibonacci diffraction lenses,” IEEE Photonics J. 5(3), 3400106 (2013).
[Crossref]

G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett. 28(12), 971–973 (2003).
[Crossref] [PubMed]

Sabatyan, A.

Seemann, R.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft X-rays with photon sieves,” Nature 414(6860), 184–188 (2001).
[Crossref] [PubMed]

Semonin, R. G.

Shi, L.

Sigarlaki, S. P.

Skibowski, M.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft X-rays with photon sieves,” Nature 414(6860), 184–188 (2001).
[Crossref] [PubMed]

Stigliani, D. J.

Sun, J. A.

J. A. Sun and A. Cai, “Archaic focusing properties of Fresnel zone plates,” J. Opt. Soc. Am. 8(1), 33–35 (1991).
[Crossref]

Tipton, M. D.

M. D. Tipton and J. E. Dowdey, “Coded aperture imaging with on-axis Fresnel zone plates,” Opt. Eng. 12(5), 166–168 (1973).
[Crossref]

Torroba, R.

V. Ferrando, A. Calatayud, P. Andrés, R. Torroba, W. D. Furlan, and J. A. Monsoriu, “Imaging properties of Kinoform Fibonacci lenses,” IEEE Photonics J. 6(1), 6500106 (2014).
[Crossref]

Urisu, T.

Wang, C.

Wang, S.

Wright, M.

T. J. Osler, M. Wright, and M. Orchard, “Theon’s ladder for any root,” Int. J. Math. Educ. Sci. Technol. 36(4), 389–398 (2005).
[Crossref]

Xie, C.

Xing, T.

Xu, F.

Xu, P.

Yu, J.

Zhang, J.

Zhao, X.

Zhou, C.

Zhu, J.

Zhu, X.

Appl. Opt. (8)

Chin. Opt. Lett. (1)

Chin. Phys. B (1)

J. Jia and C. Xie, “Phase zone photon sieve,” Chin. Phys. B 18(1), 183–188 (2009).
[Crossref]

IEEE Photonics J. (2)

V. Ferrando, A. Calatayud, P. Andrés, R. Torroba, W. D. Furlan, and J. A. Monsoriu, “Imaging properties of Kinoform Fibonacci lenses,” IEEE Photonics J. 6(1), 6500106 (2014).
[Crossref]

J. A. Monsoriu, A. Calatayud, L. Remón, W. D. Furlan, G. Saavedra, and P. Andrés, “Bifocal Fibonacci diffraction lenses,” IEEE Photonics J. 5(3), 3400106 (2013).
[Crossref]

Int. J. Math. Educ. Sci. Technol. (2)

T. J. Osler, M. Wright, and M. Orchard, “Theon’s ladder for any root,” Int. J. Math. Educ. Sci. Technol. 36(4), 389–398 (2005).
[Crossref]

C. Horgan and K. Herzinger, “A further investigation of using Theon’s ladder to find roots of quadratic equations,” Int. J. Math. Educ. Sci. Technol. 45(1), 150–158 (2014).
[Crossref]

J. Opt. Soc. Am. (2)

D. J. Stigliani, R. Mittra, and R. G. Semonin, “Resolving power of a zone plate,” J. Opt. Soc. Am. 57(5), 610–613 (1967).
[Crossref]

J. A. Sun and A. Cai, “Archaic focusing properties of Fresnel zone plates,” J. Opt. Soc. Am. 8(1), 33–35 (1991).
[Crossref]

J. Opt. Soc. Am. A (1)

Math. Mag. (1)

J. R. Ridenhour, “Ladder Approximations of Irrational Numbers,” Math. Mag. 59(2), 95–105 (1986).
[Crossref]

Nature (1)

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft X-rays with photon sieves,” Nature 414(6860), 184–188 (2001).
[Crossref] [PubMed]

Opt. Acta (Lond.) (1)

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta (Lond.) 24(4), 505–515 (1977).
[Crossref]

Opt. Eng. (1)

M. D. Tipton and J. E. Dowdey, “Coded aperture imaging with on-axis Fresnel zone plates,” Opt. Eng. 12(5), 166–168 (1973).
[Crossref]

Opt. Eng. (Bellingham) (1)

H.-H. Chung, N. M. Bradman, M. R. Davidson, and P. H. Holloway, “Dual wavelength photon sieves,” Opt. Eng. (Bellingham) 47(11), 118001 (2008).
[Crossref]

Opt. Express (4)

Opt. Lett. (5)

Other (1)

L. G. Hua, An Introduction to Number Theory (Science Press, 1979).

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Figures (9)

Fig. 1
Fig. 1 Schematic of Greek-ladder device in polar or Cartesian coordinates and their normalized intensity against the axial distance, f1 = 3.567cm, f2 = 3.439cm.
Fig. 2
Fig. 2 Axial normalized intensity of Greek-ladder ZP with square root of 1.25, 2, 3 and 4.
Fig. 3
Fig. 3 Axial intensity of Greek-ladder devices with square root of 2 against K = 0.50, 1.24 and 2.25.
Fig. 4
Fig. 4 Axial normalized intensity of Greek-ladder device with three or four foci.
Fig. 5
Fig. 5 Schematic of Greek-ladder device (the first 12 zone) with (a) (2*2)*2 array foci, (b) (2*1)*2 annular foci.
Fig. 6
Fig. 6 (a) (2*2)*2 array foci, (c) (2*1)*2 annular foci, (b) and (d) intensity contour at their own two focal planes.
Fig. 7
Fig. 7 Intensity contour at two focal planes. (a, b) 0.35 times width on y-axis. (c, d) 0.40 times width on y-axis.
Fig. 8
Fig. 8 Intensity contour at two focal planes. (a, b) stair-stepping array foci. (c, d) (3*1)*2 annular foci.
Fig. 9
Fig. 9 Intensity contour at four focal planes (a-d and e-h) with f = 2.567cm, 2.830cm, 4.584cm, 5.494cm.

Tables (4)

Tables Icon

Table 1 3-order Greek ladder with the third root of 1.25

Tables Icon

Table 2 Greek-ladder ZP with square root of 1.25, 2, 3 and 4

Tables Icon

Table 3 Greek-ladder devices with square root of 2 against K = 0.50, 1.24 and 2.25

Tables Icon

Table 4 Greek-ladder device with trifocal or four-focal distances

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

r m 2 + f 0 2 f 0 =mKλ,( m N + ,K>0 )
E 1 ( x,y,z )= 1 2π E 0 ( ξ,η,0 )t( ξ,η ) z [ exp( ikR ) R ]dξdη
r 1,j = α j ,j=1,2,...N
{ r n+1,1 = r n,1 + r n,2 +...+ r n,N r n+1,2 = r n+1,1 +( C1 ) r n,1 r n+1,3 = r n+1,2 +( C1 ) r n,2 ... r n+1,N = r n+1,N1 +( C1 ) r n,N1, C R +
γ = lim n r n , k + 1 / r n , k = C N , k = 1 , 2 , ... , N 1
{ a 1 =[ x ], R 1 =x a 1 a n =[ 1/ R n1 ], R n =1/ R n1 a n ,n2 P 0 = a 0 , P 1 = a 1 a 0 +1, P i = a i P i1 + P i2 Q 0 =1, Q 1 = a 1 , Q i = a i Q i1 + Q i2
C = [ 1 ; C 1 , 2 ; C 1 , 2 ; C 1 , 2 ; ... ]
f m = f 0 ×K [ m1 2 ]+ γ×mod( m1,2 )+1×mod( m,2 ) γ+1 , f m < f m1 <...< f 1
{ F 1 =α, G 1 =β F n+1 = G n G n+1 =m F n+1 +k F n ,( m,kR )
γ 2 mγk=0
{ F 1 =α, F 2 =β F n+1 =m F n +k F n1

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